/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 79 Evaluate using a calculator. $... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 79

Evaluate using a calculator. $$ (-2)^{-4} $$

Short Answer

Expert verified
The value of \((-2)^{-4}\) is \0.0625\.

Step by step solution

01

Understand negative exponent

Recall that a negative exponent means taking the reciprocal of the base and then raising it to the positive exponent. For example, \(a^{-n} = \frac{1}{a^n}\).
02

Apply the negative exponent rule

Apply the rule to the given expression: \((-2)^{-4}\) becomes \[\frac{1}{(-2)^4}\].
03

Calculate the positive exponent

Compute the positive exponent \((-2)^4\). Since \((-2) \times (-2) \times (-2) \times (-2) = 16\), we get \((-2)^4 = 16\).
04

Take the reciprocal

Take the reciprocal of 16 to get the final answer: \[\frac{1}{16}\].
05

Use a calculator for verification

Use a calculator to verify that \(-2^{-4} = \frac{1}{16} = 0.0625\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

reciprocal
When evaluating expressions with negative exponents, understanding the concept of a reciprocal is key. A **reciprocal** simply means flipping a number or fraction. For example, the reciprocal of 5 is \[\frac{1}{5}\]. When an exponent is negative, you take the reciprocal of the base. Thus, \((-2)^{-4}\) changes to \[\frac{1}{(-2)^4}\]. This step transforms a more complex expression into an easier one to handle.
exponentiation
Once we have taken the reciprocal, our next step is **exponentiation**, which means raising a number to a power. In this problem, we have \((-2)^4\). To calculate this, you multiply -2 by itself four times: \((-2) \times (-2) \times (-2) \times (-2) = 16\). Remember, two negative signs make a positive, so \((-2)^2 = 4\) and \((4)^2 = 16\). This process simplifies the expression significantly.
calculator verification
After completing the math manually, it's always wise to use **calculator verification**. Using a calculator ensures that a \(-2^{-4}\) indeed equals \[\frac{1}{16}\]. Simply input the original expression or the final simplified form into your calculator. For instance, entering \((-2)^{-4}\) will give you \[[0.0625]\], confirming our answer. Using a calculator can catch small mistakes and validate your solution.

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Most popular questions from this chapter

Use the fact that \(10^{3} \approx 2^{10}\) to estimate each of the following powers of \(2 .\) Then compute the power of 2 with a calculator and find the difference between the exact value and the approximation. $$ 2^{22} $$

Simplify. $$ \frac{\left(\frac{1}{2}\right)^{3}\left(\frac{2}{3}\right)^{4}}{\left(\frac{5}{6}\right)^{3}} $$

Replace \(\square\) with \(>,<,\) or \(=\) to write a true sentence. $$ 9^{7} \square 3^{13} $$

A grain of sand is placed on the first square of a chessboard, two grains on the second square, four grains on the third, eight on the fourth, and so on. Without a calculator, use scientific notation to approximate the number of grains of sand required for the 64 th square. (Hint: Use the fact that \(\left.2^{10} \approx 10^{3} \cdot\right)\)

Replace \(\square\) with \(>,<,\) or \(=\) to write a true sentence. $$ 4^{2} \square 4^{3} $$
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